It all depends on what era you;re in.
Assuming you're at the Rifle/Gren/Galleon stage, here's the deal
Build 3 Galleons for each coastal city, and 2 for each inland city. Bring along units with Drill and Combat promotions. Rifles and Cannons will be the main attacking troops, Grenadiers for pillaging and mopping up remaining units, and City Garrison-equipped units for fortification. Send 3 Galleons to each city, and more if the city is close to an inland one. Land a tile away from the city, and use the Galleons to blockade. Bring along 1 Spy for each city, if possible, or 1 Spy for 2 cities. Use the Spy to send the city into anarchy, then attack. If you don't have a Spy for the city, or too few Espionage Points, use the Cannons to bombard. Either way, attack with the Rifles, then the Grens. Remember to send a few Galleons running back and forth to shuttle more and more units. After the city is captured, use the blockading Galleon to join the shuttle fleet. Attack the inland like you would with any other Civ.
Assuming you're at the Modern Era
Build a massive fleet. 45 Transports is good. Build a few Carriers stocked with Bombers and Fighters. Mass Marines, MGs, and Artillery, as well as Gunships. Tanks can be used, but its optional. Use a few Destroyers and Battleships to guard the convoy. Line the Transports near the cities. Use the Artillery to destroy defensive bonuses. Since Marines have Amphibious, attack directly from the Transport. Use the guarding Destroyers to blockade and put some on Sentry to protect. Again, use the Transports to shuttle troops. Fortify the MGs on the captured city, and use the Gunships to pillage. Rinse and repeat for every city.
Way more than you really need to start the war. Brute force is way too expensive for an overseas war. You are talking about almost 250 units if you include the ships. Even if you have 50 of them in the new cities as garrisons, you lose 200gpt. That is an EXTRA 200gpt in addition to the unit maintnance to begin with. Overseas wars are won strategically not tactically. And the key to winning a war like that is attrition. I must say i am the master of the passive war of attrition since I play lots of AggAi games and tend to be a builder by nature. So by the time I finally decide to start a war overseas, the Ai has huge stockpiles of troops.
What you will need
Ships...about ten-15 frigates/destroyers/battleships. These are just for the invasion fleet....continue to build more in coastal cities for the duration of the war at a pace equal to your naval losses. Twenty-five galleons or Nineteen transports. One caravel.
Ground troops. You will be taking a total of 75 for the initial assault. Since that is the maximum your ships can carry. 10 cannons with the maximum City raider promotions. 15 marines if they are available. If not then 15 of your best assault troops. One unit with medic promotion. Preferably a scout or explorer. If you have a medic three mash unit then bring him. That leaves you 49 units. Bring 30MGs and the rest the best defenders you have Infantry would be best, You can tech to them while you build the MG's. If you can build CG3 units with a promotion left over use it for Guerrilla1. If tanks and gunships are in play for the AI make it 25MGS 10Saminfantry with intercept promotions and CG1, 10Antitank with maximumCG, and nine marines with max CG.
air units....fighters atleast 12 and of course 4 carriers to bring them.
What you will do. You should have the caravel snoop about to find a coastal city on a hill. Yes hill cities are tougher to take but they are the best for the stack-eater strat. When not at war, the Ai tends to mass the troops in a few cities and have 3-4 defenders in the rest. Find one on a hill not to far from the best cities. You want your beachhead someplace conveniently located to the best targets. Position your assault fleet within one turn from that city.
if you have a spy in the territory then send him to the city. If not, no big deal.
Step one: declare war (very important step, do not forget it)
Step two: Move ships next to city to be taken.
Step three: Use the ships to reduce the targets defenses. If that fails then use the spy. Try the ships first to save EPs and so your spy stays on that continent.
Step four: suicide all your cannons. They will not have amphibious, they will die, they will be missed, life goes on...just not for them.
Step five: Attack with your assault troops. Even if they are not amphibious (either through promotions or being marines) 15 will kill 3-4 cannon weakened defenders. Many will die in the process, they will be missed, life goes on....just not for them.
Step six move all your units into the city and fortify them.
Step seven send the ships back for more defenders.
step Eight, watch the Ai send waves and waves of troops to try and take the city back.
You will be absolutely amazed at how many troops these defenders will kill. Playing a recent game as Charlemagne I had a city able to produce level 5 units right out of the box. They were CG3 Guerilla2 drill1 defenders. That is plus 125% and a first strike. The infatnry were 45 against any nonpinch promoted unit and strength 40 vs pinch units. Of course the Ai attacked with waves of cannons. Which the MGs butchered. And the MGs suffer no collateral damage at all. The MGs can't get guerilla promotions so they were CG3 Drill3. And a CG3 MG is strength 40.5 vs non pinch gunpowder units and strength 36 vs pinch units.
Step nine, go about your business back on the mainland and pretty much ignore the war overseas.Using naval units to kill incoming fleets. They have to build a lot more units than you do since you get to kill the troops on board without having to fight them. You don't pay the added costs for troops since they are in a city of yours and only troops outside your cultural boundries cost extra during a war. the Ai switches to expensive wartime techs that deplete the treasury and slow research. They lose the benefits of the peacetime civics (Beauracracy and free religion or OR and even worse if they go into nationhood). If they don't have jails and/or Mt rushmore then war weariness absolutely cripple them because the assault losses on your little beachead city count as foreigh territory. Just send over the occasional pile of fresh defenders from time to time. and watch the AI sink into techbackwardness with little production because his people are war weary to the point noone is working. I have actually watched cities shrinking because noone will work the food. Then i poison the well....cuz that's how i am. Eventually I might go over and beat him up and take some more cities, but that is a pain in the butt. I'd rather build some wonders. I once had a never ending war that lasted like that until I had Mech infantry. I loaded up about 30 of them and that was that. the Ai never gives up trying to take that city. After5turns the defenders get yet another 25% defense from fortifying. I suppose that once his troop stockpile is wiped out you could wander over with more troops and invade. His arty/cannons are pretty much toast and his cities are barely defended.
) If war weariness is killing your economy, get Fascism and build Mt.Rushmore and jails, and run police state if you are able.
You need a lot of soldiers for a successful overseas invasion. I mean a LOT. Those soldiers will face the full force of a continent, on just a handful of tiles of land. If you are unable to comfortably support your attack and have to drastically lower your science rate, you probably have no business invading someone overseas anyway.
